1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Aloiza [94]
3 years ago
15

A room has eight switches, each of which controls a different light. Initially, exactly five of the lights are on. Three people

enter the room, one after the other. Each person independently flips one switch at random and then exits the room. What is the probability that after the third person has exited the room, exactly six of the lights are on? Express your answer as a common fraction.
Mathematics
1 answer:
aivan3 [116]3 years ago
8 0

Answer:

The probability that 3 lights are on after the third person exited the room is 39/128

Step-by-step explanation:

In order for 6 switches to be on at the end, we need exactly 2 people turning on a switch and the other one turning one off. There are 3 possibilities:

  • The first two persons turn the switch in and the last one turns it off
  • The first and last person turn the switch in and the middle one tunrns it off
  • The first person turns the switch off and the 2 remaining turn the switch in

Note that after turning off one switch one more switch will be available to be switched in and one less will be available to be switch off. The contrary happens when someone turns in a switch.

Lets calculate the probability for the first scenario. The probability for the first person to turn the switch in is 3/8, because there are 3 lights off. For the second person, there will be only 2 lights off, thus, the probability for him or her to turn the switch in is only 2/8, leaving only 1 light off and 7 on. The third person will have, as a consecuence, a probability of 7/8 to turn off one of the 7 switches. This gives us a probability of 3/8 * 2/8 * 7/8 = 21/256 for the first scenario.

For the second scenario we will have a probability of 3/8 for the first person, a probability of 6/8 for the second one (he has to turn a switch off this time), and a probability, again, of 3/8 for the third one, giving us a probability of 3/8*6/8*3/8 = 27/256 for the second scenario.

For the third scenario, the first person has to turn off the switch, and it has a probability of 5/8 of doing so. The second person will have 4 switches to turn on, so it has a probability of 4/8 = 1/2, and the third person will have one switch less, thus, a probability of 3/8 of turning a switch on. Therefore, the probability of the third scenario is 5/8*1/2*3/8 = 15/128 = 30/256

By summing all the three disjoint scenarios, the probability that six lights are on is 21/256+27/256+30/256 = 78/256 = 39/128.

You might be interested in
An are whose measure is exactly 180°​
Elenna [48]

Answer:

<u><em>Mmeasure is exactly 180°​</em></u>

semi-circle because the measure of a whole circle is 360°.

<u><em> Arc :</em></u>

The arc of a circle consists of two points on the circle and all of the points on the circle that lie between those two points. It's like a segment that was wrapped partway around a circle. An arc is measured not by its length (although it can be, of course) but most often by the measure of the angle whose vertex is the center of the circle and whose rays intercept the endpoints of the arc. Hence an arc can be anywhere from 0 to 360 degrees.

5 0
3 years ago
Write the quadratic function in vertex form.
melisa1 [442]
Complete the square
isolate x terms
y=(x^2-14x)+53
take 1/2 of -14 and square it ((-7)^2=49
add plus and minus of that inside the parntehasees

y=(x^2-14x+49-49)+53
factor perfect suqrae
y=((x-7)^2-49)+53
get rid of parnthasees
y=(x-7)^2-49+53
y=(x-7)^2+4

C isi answer
6 0
3 years ago
Create a word problem that could be used to describe 8x=48
enyata [817]

Answer:

Alex goes to the shop and buys 4 times a number bananas (X). When he counts how many bananas he has he counts 48. How much is the number x?

7 0
3 years ago
Choose the equation that cannot be used to find the
Sphinxa [80]

Answer:m<b=m<d+125 degrees

Step-by-step explanation:

3 0
3 years ago
Annabelle invested $5,900 in an account paying an interest rate of 2% compounded continuously. Assuming no deposits or withdrawa
igor_vitrenko [27]

Answer:

assuming No Deposits Or Withdrawals Are Made, How Long Would It Take, To The Nearest Year, For The Value Of The Account To Reach $7,440? ... invested $5,900 in an account paying an interest rate of 2% compounded continuously

4 0
3 years ago
Read 2 more answers
Other questions:
  • Which one is greater 3/8 or 2/6
    15·1 answer
  • Order the numbers 0.2, 1/5, 0.02, 0.022 from least to greatest.
    12·1 answer
  • 13 cans of paint with 5/6 liters of paint in each were used to paint 4 identical cars. How can you calculate the amount of paint
    13·1 answer
  • Anyone feel like helping me??
    8·1 answer
  • Marnie conducted a survey of the cats in four shelters and wrote the following observations:
    5·2 answers
  • A line segment connecting any two points on a circle
    13·2 answers
  • An element with a mass of 890 grams decays by 19.7% per minute. To the nearest tenth of a minute, how long will it be until ther
    14·2 answers
  • Please help me on this
    13·1 answer
  • Please Help! Solve the equation for x.
    8·1 answer
  • Hillsdale's zoo has two elephants. The male elephant weighs 9 7/12 tons and the female elephant weighs 3 5/6 tons. How much more
    12·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!